bei48482_FM

Wave Properties of Particles 111

The de Broglie wavelength of a particle of momentum p is hp and the

corresponding wave number is

k 

In terms of wave number the particle’s momentum is therefore

p 

Hence an uncertainty kin the wave number of the de Broglie waves associated with the

particle results in an uncertainty pin the particle’s momentum according to the formula

p 

Since xk ^12 , k 1 (2x) and

x p (3.21)

This equation states that the product of the uncertainty xin the position of an ob-

ject at some instant and the uncertainty pin its momentum component in the xdi-

rection at the same instant is equal to or greater than h 4 .

If we arrange matters so that xis small, corresponding to a narrow wave group,

then pwill be large. If we reduce pin some way, a broad wave group is inevitable

and xwill be large.

h



4 

Uncertainty

principle

h k



2 

hk



2 

2 p



h

2 





Werner Heisenberg (1901–1976)

was born in Duisberg, Germany,

and studied theoretical physics at

Munich, where he also became an

enthusiastic skier and moun-

taineer. At Göttingen in 1924 as an

assistant to Max Born, Heisenberg

became uneasy about mechanical

models of the atom: “Any picture

of the atom that our imagination

is able to invent is for that very

reason defective,” he later remarked. Instead he conceived an

abstract approach using matrix algebra. In 1925, together with

Born and Pascual Jordan, Heisenberg developed this approach

into a consistent theory of quantum mechanics, but it was so

difficult to understand and apply that it had very little impact

on physics at the time. Schrödinger’s wave formulation of

quantum mechanics the following year was much more suc-

cessful; Schrödinger and others soon showed that the wave and

matrix versions of quantum mechanics were mathematically

equivalent.

In 1927, working at Bohr’s institute in Copenhagen, Heisen-

berg developed a suggestion by Wolfgang Pauli into the uncer-

tainty principle. Heisenberg initially felt that this principle was

a consequence of the disturbances inevitably produced by any

measuring process. Bohr, on the other hand, thought that the

basic cause of the uncertainties was the wave-particle duality,

so that they were built into the natural world rather than solely

the result of measurement. After much argument Heisenberg

came around to Bohr’s view. (Einstein, always skeptical about

quantum mechanics, said after a lecture by Heisenberg on the

uncertainty principle: “Marvelous, what ideas the young people

have these days. But I don’t believe a word of it.”) Heisenberg

received the Nobel Prize in 1932.

Heisenberg was one of the very few distinguished scientists

to remain in Germany during the Nazi period. In World War II

he led research there on atomic weapons, but little progress had

been made by the war’s end. Exactly why remains unclear, al-

though there is no evidence that Heisenberg, as he later claimed,

had moral qualms about creating such weapons and more or

less deliberately dragged his feet. Heisenberg recognized early

that “an explosive of unimaginable consequences” could be de-

veloped, and he and his group should have been able to have

gotten farther than they did. In fact, alarmed by the news that

Heisenberg was working on an atomic bomb, the U.S. govern-

ment sent the former Boston Red Sox catcher Moe Berg to shoot

Heisenberg during a lecture in neutral Switzerland in 1944.

Berg, sitting in the second row, found himself uncertain from

Heisenberg’s remarks about how advanced the German program

was, and kept his gun in his pocket.

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